Our use of the abbreviation \(\textrm {elpd}\)  follows the example of [1]}.
Under mild assumptions about \(\mathcal {P}\)  and \(\Pi ,\)  \(\textrm {lpd}(\mathcal {D}^{\star };\Pi ) \rightarrow \textrm {elpd}(\Pi )\)  as the number of testing points \(N^{\star }\)  diverges.
Expected log-predictive density is closely related to the Kullback–Leibler divergence:
\(\textrm {KL}\left(\mathcal {P}(x^{\star }) \mathbin {\Vert }\Pi (x^{\star }\vert \mathcal {D}) \right) = \int {p(x^{\star })\log p(x^{\star })dx^{\star }} - \textrm {elpd}(\Pi ).\) 
